CLASS 8 · GANITA PRAKASH PART I · CHAPTER 4
Original short and explained answers for quadrilateral definitions, constructions, angle and diagonal properties, and all printed Figure it Out questions.
60 worked answers · short steps first, explanations when you need them
Textbook pages 83–93
Question carpenter-diagonal
A rectangle’s diagonals are equal.
So the second diagonal must match the first.
8 cm.
Question rectangle-intersection
In a rectangle, the diagonals bisect each other.
So they meet at the midpoint of each strip.
Join them at their common midpoint.
Question rectangle-equal-bisect
Diagonals that bisect each other imply a parallelogram.
A parallelogram with equal diagonals is a rectangle.
It is a rectangle (a square is a special rectangle).
Question rectangle-diagonal-proof
In △ABC and △BAD: AB is common; ∠ABC=∠BAD=90°; BC=AD (opposite sides of a rectangle).
The triangles are congruent by SAS.
Their corresponding diagonals are equal: AC=BD.
AC=BD.
Question rectangle-bisects-proof
A rectangle is a parallelogram.
The diagonals of a parallelogram bisect one another.
So O is the midpoint of both diagonals: OA=OC and OB=OD.
OA=OC and OB=OD.
Question rectangle-angle-figure-i
∠2=90°−30°=60°; matching isosceles triangles give ∠3=∠7=60° and ∠4=∠5=30°.
Then ∠6=60° and ∠8=30°.
At the diagonal intersection: ∠9=120°, ∠10=60°, ∠11=120°, ∠12=60°.
∠2=60°, ∠3=60°, ∠4=30°, ∠5=30°, ∠6=60°, ∠7=60°, ∠8=30°, ∠9=120°, ∠10=60°, ∠11=120°, ∠12=60°.
Question rectangle-angle-figure-ii
∠11=110° (vertically opposite); ∠10=∠12=70° (linear pairs).
The isosceles triangle gives ∠1=∠8=(180°−110°)÷2=35°.
Each corner is 90°, so ∠2=∠3=∠6=∠7=55° and ∠4=∠5=35°.
∠1=35°, ∠2=55°, ∠3=55°, ∠4=35°, ∠5=35°, ∠6=55°, ∠7=55°, ∠8=35°, ∠10=70°, ∠11=110°, ∠12=70°.
Question equal-diagonals-arbitrary-angle
Draw one 8 cm diagonal and mark its midpoint O.
Through O draw the other diagonal at the chosen angle; mark 4 cm on each side of O.
Join the four endpoints in order. Repeat for each angle.
Each construction is a rectangle; for 90° it is a square.
Question perpendicular-diameters
All four vertices lie on the circle, and the diagonals are the two diameters.
The diagonals are equal and bisect one another at the centre; they also meet at 90°.
APML is a square.
Question sticks-right-angle
Cross the equal sticks at their midpoints, making them the two diagonals of a quadrilateral.
Adjust the crossing until the four outer endpoints have equal distances around the boundary; tie the thread around those endpoints.
The resulting quadrilateral has equal diagonals that bisect each other and is a rectangle; equal adjacent sides make it a square, whose corners are 90°.
Use the sticks as equal perpendicular diagonals of a square; its corners give exact right angles.
Question parallel-equal-not-rectangle
Those conditions define a parallelogram.
A slanted parallelogram can have no right angles.
No. A non-rectangular parallelogram is a counterexample.
Question Rectangle definition
It is a quadrilateral.
All four angles are 90°; its opposite sides have equal lengths.
A rectangle is a quadrilateral with four right angles and equal opposite sides.
Question Rectangle from diagonal strips
The diagonals of a rectangle are equal, so the second strip is 8 cm.
The diagonals bisect each other, so join them at their midpoints, 4 cm from each end.
Use a second 8 cm strip and join both strips at their midpoints.
Question Why equal bisecting diagonals make a rectangle
Let the diagonals cross at O with one angle x. Opposite angles are x and 180°−x.
Equal diagonals give isosceles triangles; their base angles add to 90°−x/2 and x/2.
Each corner angle is (90°−x/2)+(x/2)=90°.
All four corner angles are 90°, so the quadrilateral is a rectangle.
Question Rectangle with four right angles
Draw a diagonal to split the quadrilateral into two triangles.
Each triangle has a right angle, a shared diagonal, and one matching right angle; the remaining angles are equal because each completes 90°.
The triangles are congruent by AAS, so their corresponding opposite sides are equal.
Its opposite sides are equal, so it is a rectangle.
Question Square properties
All four sides are equal; all four angles are 90°.
Opposite sides are parallel. The diagonals are equal, bisect each other at 90°, and bisect the corner angles.
A square has four equal sides, four right angles, parallel opposite sides, and diagonals that are equal, perpendicular bisectors of each other and of the angles.
Question Square diagonal construction, 8 cm
Draw an 8 cm segment and construct its perpendicular bisector.
Mark 4 cm on each side of the midpoint along the perpendicular line.
Join the four endpoints in order.
The endpoints form a square.
Question Square diagonal angle halves
A diagonal makes an isosceles right triangle because two sides of the square are equal.
The two remaining angles are equal and sum to 90°.
Each is 45°.
Each square diagonal bisects a corner into two 45° angles.
Textbook pages 94–96
Question angle-sum-proof
Draw a diagonal to split it into two triangles.
Each triangle has angle sum 180°.
180°+180°=360°.
The interior angles of every quadrilateral sum to 360°.
Question three-right-angles
Three right angles total 270°.
The fourth angle is 360°−270°=90°.
No. The fourth angle must also be 90°.
Question concave-angle-sum
Draw a diagonal that divides the concave shape into two triangles (or split its reflex angle into the corresponding triangle angles).
The two triangle sums total 180°+180°.
Yes, the interior angles still total 360°.
Textbook pages 96–103
Question quadrilateral-remaining-angles-i
Opposite angles in a parallelogram are equal.
Adjacent angles are supplementary: 180°−40°=140°.
The opposite angle is 40°; the two adjacent angles are 140° each.
Question quadrilateral-remaining-angles-ii
The adjacent angle is 180°−110°=70°.
Opposite angles match.
The remaining angles are 70°, 110° and 70° in order.
Question quadrilateral-remaining-angles-iii
The diagonal splits a 60° interior angle into two 30° parts.
Adjacent parallelogram angles total 180°.
The labelled angles are 30°, 60°, 30°, 120° and 120° as shown in the answer key: ∠XVU=∠XVW=30°, ∠WXU=∠UVW=60°, ∠UXV=∠WXV=30°, and ∠U=∠W=120°.
Question quadrilateral-remaining-angles-iv
The diagonal gives the marked 20° corresponding angles.
Adjacent angles on a straight line or in the quadrilateral fill the remaining parts.
∠OEI=20°, ∠AOE=20°, ∠EOI=20°, and ∠A=∠I=140°.
Question construct-parallelogram-diagonals
Draw the 7 cm diagonal and mark its midpoint O.
At O construct a 140° line. Mark 2.5 cm on each side for the second diagonal.
Join the four endpoints in order.
The constructed quadrilateral is the required parallelogram.
Question construct-rhombus-diagonals
Draw the 5 cm diagonal and mark midpoint O.
Draw a perpendicular through O and mark 2 cm on each side.
Join the four endpoints.
The resulting quadrilateral is a rhombus.
Question rhombus-perpendicular-diagonals
Let diagonals AC and BD meet at O. A rhombus has equal sides, so AB=BC; also AO=OC (its diagonals bisect).
In △AOB and △COB, AB=CB, AO=CO, and BO is common; the triangles are congruent by SSS.
Thus ∠AOB=∠BOC. They form a straight angle, so each is 90°.
The diagonals of a rhombus intersect at right angles.
Question Parallelogram construction and measures
Opposite sides are equal: the remaining sides are 4 cm and 5 cm.
Adjacent angles are supplementary: 180°−30°=150°.
Opposite angles are equal.
The sides are 4 cm, 5 cm, 4 cm, 5 cm; the angles are 30°, 150°, 30°, 150°.
Question Parallelogram opposite angles
An adjacent angle is 180°−x° because same-side interior angles between parallel lines are supplementary.
The opposite angle is x°; the other adjacent angle is 180°−x°.
The four angles are x°, 180°−x°, x°, 180°−x° in order.
Question Parallelogram opposite sides proof
Draw one diagonal. It creates two triangles.
Alternate interior angles are equal because opposite sides are parallel; the diagonal is common.
The triangles are congruent by ASA, so corresponding opposite sides are equal.
Both pairs of opposite sides are equal.
Question Parallelogram diagonal properties
They do not have to be equal; a slanted parallelogram can have unequal diagonals.
They do bisect each other: the intersection is the midpoint of each diagonal.
The diagonals need not be equal, but they always bisect one another.
Question Rhombus with a 50° angle
Opposite angles in a rhombus are equal.
Adjacent angles sum to 180°: 180°−50°=130°.
The angles are 50°, 130°, 50°, 130° in order.
Question Rhombus diagonals
They bisect one another.
They meet at 90°.
Each diagonal bisects the two opposite vertex angles it connects.
The diagonals bisect each other perpendicularly and bisect the rhombus’s angles.
Textbook pages 103–108
Question two-equilateral-triangles
Each triangle angle is 60°.
The four outside sides are all 4 cm.
At the two shared-edge endpoints, outer angles are 60°+60°=120°; the other two remain 60°.
The outer shape is a rhombus: all sides are 4 cm and its angles alternate 60° and 120°.
Question construct-kite-diagonals
Draw the 6 cm diagonal.
Construct its perpendicular bisector.
Mark 4 cm each way on this perpendicular for the 8 cm diagonal, then join the endpoints.
The resulting quadrilateral is a kite (in this symmetric construction, it is also a rhombus).
Question trapezium-angles-a
For parallel bases, consecutive interior angles on a leg add to 180°.
In the marked isosceles trapezium, the base angles are equal; subtract from 180° to find the adjacent angle.
The answer-key values are ∠S=45° in the first diagram, ∠R=75° in the second, and ∠A=∠B=80° for the isosceles trapezium.
Question kite-parallelogram
A parallelogram has opposite sides equal. For it also to have adjacent equal sides, all four sides must be equal.
A parallelogram with four equal sides is a rhombus.
Rhombuses, including squares.
Question kite-rectangle-definition
A square has two pairs of equal adjacent sides and four right angles, so it is both a kite and a rectangle.
A non-square rectangle does not have adjacent equal sides; many kites do not have all four sides equal.
A square is both a kite and a rectangle. Not every kite is a rhombus; every rhombus is a kite.
Question rectangle-angle-iod
Use the right angles of each rectangle to compare the angles made by their diagonals and parallel sides.
The corresponding angle at O matches the given 30°.
∠IOD=30°.
Question construct-square-diagonal
Draw a 6 cm segment and construct its perpendicular bisector with a compass.
From the midpoint, mark 3 cm each way on the perpendicular.
Join the four endpoints.
The four sides are equal and adjacent sides are perpendicular, so the figure is a square.
Question midpoint-square
Each small right triangle has legs x/2 and x/2.
By Pythagoras, each inner side is √((x/2)²+(x/2)²)=x/√2.
The four sides are equal and each corner is 90°.
UVWX is a square with side x/√2.
Question four-equal-sides-one-right
Four equal sides make a rhombus.
Opposite angles in a rhombus are equal and adjacent angles are supplementary.
One 90° angle forces every angle to be 90°.
Yes, it is a square.
Question opposite-sides-equal
Draw a diagonal.
The two resulting triangles have three equal corresponding sides, so they are congruent by SSS.
Their alternate interior angles are equal, so both pairs of opposite sides are parallel.
It is a parallelogram.
Question concave-quadrilateral-sum
Draw a diagonal to divide the shape into two triangles.
Each triangle has angle sum 180°.
Yes. Its interior angles add to 360°.
Question tf-equal-bisect
Check the defining properties and use the angle sum or a counterexample.
False. It must be a rectangle, but its diagonals need not be perpendicular; a non-square rectangle is a counterexample.
Question tf-three-right
Check the defining properties and use the angle sum or a counterexample.
True. The fourth angle is 360°−270°=90°, so all four angles are right.
Question tf-bisect-parallelogram
Check the defining properties and use the angle sum or a counterexample.
True. This is the converse of the parallelogram diagonal property.
Question tf-perpendicular-rhombus
Check the defining properties and use the angle sum or a counterexample.
False. A kite can have perpendicular diagonals without all four sides being equal.
Question tf-opposite-angles
Check the defining properties and use the angle sum or a counterexample.
True. Both pairs of opposite angles match; using the 360° sum shows adjacent angles are supplementary, making both side pairs parallel.
Question tf-all-angles-equal
Check the defining properties and use the angle sum or a counterexample.
True. The angles total 360°, so each is 90°.
Question tf-isosceles-trapezium-parallelogram
Check the defining properties and use the angle sum or a counterexample.
False. Only one pair of opposite sides is parallel in a general trapezium; the other pair need not be parallel.
Question What makes a quadrilateral
A quadrilateral is a closed polygon made from exactly four straight line segments.
A curved boundary or a shape that is not a single closed four-sided polygon does not qualify.
A quadrilateral must be closed and have four straight sides.
Question Geoboard with equal perpendicular diagonals
Equal diagonals that bisect each other make a rectangle.
If they are also perpendicular, all sides are equal: it is a square.
Lengthening just one diagonal makes the diagonals unequal but still perpendicular and bisecting; that gives a rhombus.
First shape: a square. After extending one diagonal equally at both ends: a rhombus.
Question Join two equilateral triangles of side 8 cm
Each outside side is one triangle side, so all four are 8 cm.
The two 60° angles at each joined endpoint combine to 120°; the other two outside angles remain 60°.
The outside shape is a rhombus with four 8 cm sides and angles 120°, 60°, 120°, 60°.
Question Join two isosceles triangles
Join the triangles along a matching 6 cm side, with the other vertices on opposite sides of that shared side.
The outside boundary has two pairs of equal neighboring sides: 8 cm, 8 cm, 8 cm, 8 cm in this arrangement.
One valid arrangement is a kite; the shared 6 cm side becomes an internal diagonal.
Question Join two scalene triangles
Join the triangles along one matching side, such as the 12 cm side, with the triangles on opposite sides of it.
The outside boundary has two adjacent 6 cm sides and two adjacent 9 cm sides.
This arrangement makes a kite with side sequence 6, 9, 9, 6 cm.
Question Kite diagonal properties
Triangles ABD and CBD have AB=BC, AD=CD and shared side BD, so they are congruent by SSS.
Thus BD bisects ∠B and ∠D, and the intersection O satisfies AO=OC.
The equal adjacent angles at O lie on a straight line, so BD⊥AC.
The symmetry diagonal BD bisects the kite’s two opposite angles and the other diagonal at 90°.
Question Trapezium angles
A non-parallel side is a transversal of the parallel sides.
Same-side interior angles are supplementary.
∠P+∠S=180° and ∠Q+∠R=180° (with the usual cyclic labelling).
Question Isosceles trapezium base angles
The non-parallel sides are equal, making the two end triangles formed by perpendiculars congruent.
Therefore the two angles at each parallel base are equal.
Each pair of base angles is equal; angles on the same leg still add to 180°.